Groups in Geometry and Topology

For an orientable surface $S_g$, characteristic classes of $S_g$-bundles are given by the cohomology of the classifying space of $\text{Diff}^+(S_g)$, the group of orientation preserving diffeomorphisms of $S_g$. Moreover, for $g \geq 2$ it is well known that $H^*( B \text{Diff}^+ (S_g)) = H^*(\Gamma_g)$, where $\Gamma_g$ is the mapping class group of $S_g$. In this work we look at the analogues for non-orientable surfaces. For instance, in the case of the Klein bottle $K$ we show the cohomology of $B\text{Diff}(K)$ is given by the cohomology of $\mathbb Z/2\times \mathbb Z/2$ with twisted coefficients in $H^*(\mathbb C \text{P}^\infty)$ and determine its homotopy type. We also provide concrete Eilenberg-MacLane spaces $K(\pi,1)$ for the mapping class groups with marked points $\Gamma^q(\mathbb R \text{P}^2)$ and $\Gamma^q(K)$ and use them to express the cohomology of these groups in terms of the cohomology of configuration spaces. Further relations to mapping spaces are given. This is joint work with C. Hidber and M. Maldonado.

I will discuss finite group actions on integral or rational homology 3-spheres. The main examples for this talk are the Brieskorn integral homology 3-spheres $M(p,q,r)$ arising from isolated singularities, which bound smooth 4-manifolds with definite intersection forms. In addition, there are special infinite families of Brieskorn homology 3-spheres which can be realized as boundaries of smooth contractible 4-manifolds. We ask whether the free periodic actions on Brieskorn spheres extend to smooth actions with isolated fixed points on one of these associated 4-manifolds.

An equivariant infinite loop space machine is a functor that constructs genuine equivariant spectra out of simpler categorical or space level data. In the late 80's Lewis-May-Steinberger and Shimakawa developed generalizations of the operadic approach and the Gamma-space approach respectively. In this talk I will describe work in progress that aims to understand these machines conceptually, relate them to each other, and develop new machines that are more suitable for certain kinds of input. This work is joint with Anna Marie Bohmann, Bert Guillou, Peter May and Mona Merling.

We discuss homotopy group actions and invariants obtained using group cohomology.

By studying the fixed points of G-equivariant $n$-dimensional complex vector bundles, Gomez and Uribe give a decomposition formula of equivariant complex vector bundles in terms of twisted equivariant complex vector bundles of smaller dimension. We use this decomposition to describe a decomposition of the equivariant unitary bordism groups for adjacent families of subgroups.

Let $G$ be a group, $\mathcal{F}$ a family of subgroups of $G$, $\mathcal{C}$ a $\mathbb{Z}$-linear category equipped with a linear $G$-action, $\mathcal{C}\rtimes G$ the crossed product, and $E$ a functor from $\mathbb{Z}$-linear categories to spectra. The $G$-equivariant $E$-homology with coefficients in $\mathcal{C}$ associates a spectrum $H^G(X,E(\mathcal{C}))$ to every $G$-space $X$, in such a way that there is a weak equivalence $E(\mathcal{C}\rtimes H)\overset{\sim}\to H^G(G/H,E(\mathcal{C}))$. The isomorphism conjecture for the triple $(G,\mathcal{F},E)$ with coefficients in $\mathcal{C}$ says that if $\mathcal{E}_{\mathcal{F}}G$ is a classifying space for $G$ with respect to $\mathcal{F}$, then the map $\mathcal{E}_{\mathcal{F}}G\to G/G=pt$ induces a weak equivalence \[ H^G(\mathcal{E}_{\mathcal{F}}G,E(\mathcal{C}))\overset{\sim}\to E(\mathcal{C}\rtimes G) \] When $\mathcal{C}$ is a $C^*$-algebra, $\mathcal{F}=\mathcal{F}in$ is the family of finite subgroups $E=K^{\mathrm{top}}$ and $\rtimes$ is the reduced $C^*$-crossed product, this is the Baum-Connes conjecture with coefficients $\mathcal{C}$, and the groups in the left hand side \[ H_*^G(\mathcal{E}_{\mathcal{F}in}G,K^{\mathrm{top}}(\mathcal{C}))=KK_*^G(\mathcal{E}_{\mathcal{F}in}G, \mathcal{C})\] are the Kasparov equivariant bivariant $K$-theory groups. In the talk I will report about current progress on an old project that aims to describe the left hand side of the isomorphism conjecture for Weibel's homotopy algebraic $K$-theory in terms of Ellis' $G$-equivariant bivariant algebraic $K$-theory.

We will describe rigidity properties for manifolds that are built as graphs of groups with specific geometric properties.

Localized mirror functors provide explicit geometric mechanisms relating Fukaya category and the category of matrix factorizations. In the case of orbifold spheres with three orbifold points, this proves homological mirror symmetry from orbifold spheres to mirror Landau-Ginzburg models. In the construction, finite group actions play an important role and we explain its motivation and implications. This is based on joint works with Hansol Hong and Siu-Cheong Lau.

We will discuss some recent applications of gauged maps into quantum K-theory computations for GIT quotients of a projective variety by a reductive group. This is joint work with C. Woodward and some parts with P. Solis.

Two groups are called isocategorical over a field k if their respective categories of k-linear representations are monoidally equivalent. We classify isocategorical groups over arbitrary fields, extending the earlier classification of Etingof-Gelaki and Davydov for algebraically closed fields. In order to construct concrete examples of isocategorical groups a new variant of the Weil representation associated to isocategorical groups is defined. We construct examples of non-isomorphic isocategorical groups over any field of characteristic different from two and rational Weil representations associated to symplectic spaces over finite fields of characteristic two.

The remarkable work of Edward Witten on the study of topological aspects of \emph{quantum field theory}, inspired the concept of a \emph{topological quantum field theory} (shortly TQFT) by Michael Atiyah and Graeme Segal. Michael Atiyah defines a TQFT in terms of axioms motivated by the physical properties in quantum field theory; while Graeme Segal defines a TQFT as a symmetric monoidal functor from the cobordism category to the category of complex vector spaces. Initially, we are going to prove that this two definitions are exactly the same. In this talk we propose a method to construct TQFT in high dimensions, motivated by the work of Markus Banagl, in order to detect exotic structures on spheres.

We describe how the notion of diffeology can apply to Mulase constructions in the KP hierarchy. The existence and the uniqueness of the solution is here re-interpreted with rigorous differential geometric constructions, by two different approaches. The first one is based on the algebaric approach of Reyman and Semenov-tian-Shansky, and the second one describes the solutions of the initial KP hierarchy as the infinite jet of a global section of a principal bundle. These two approaches show very naturally that the solutions depend smoothly of the initials. We finish with three perspectives: the relationship between the KP hierarchy and the KP-I equation, the possibility to consider equations envolving non smooth functions, and the extension of the second method to other Lax-type equations.

I'll detail a procedure to describe T-duality and mirror symmetry of certain nilmanifolds possibly endowed with gerbes at the level of non-BPS states. This will be realized as an isomorphism of two Vertex algebra-like objects naturally attached to these nilmanifolds. To any Courant algebroid over a manifold M one can attach a sheaf $\Omega$ of vertex algebras over M. When M is not simply connected, for any conjugacy class $\omega \in \pi^1(M)$ on the fundamental group of M we have a associated an $\Omega$ module $M_{[\omega]}$. Taking global sections and summing over all "winding" classes, the space $\oplus \Gamma(M, M_{[\omega]})$ carries a structure that reminds of that of a vertex algebra and turns out to be naturally isomorphic with the corresponding structure on the mirror dual manifold. The talk is based on previous work joint with M. Aldi and further generalized by J. Villareal Montoya in his 2017 Ph.D. thesis.

I will discuss a symplectic version of the usual theory of bundle gerbes. This notion of gerbe is relevant, e.g, to understand non-commutative integrable systems or for the theory of Poisson manifolds of compact types. The talk is based on joint work with Marius Crainic and David Martinez Torres.